A classical motivating example for the exponential eigenvalue decay
is the squared exponential kernel, where an
analytical form of the decay can be derived for normally
distributed data [1]}. In particular,
assuming \(x \sim (0,\eta ^2)\) , and using the kernel
function \(k(x,y) =\exp \big (-\frac{(x-y)^2}{2l^2}\big )\)  in one dimension, the
eigenvalues satisfy \(\lambda _k \le C\gamma ^k\)  for a general constant \(C\)  independent of \(k\) ,
where
\(\gamma = \frac{2\eta ^2}{l^2 + 2\eta ^2 + \sqrt{l^2 + 2\eta ^2}}.\) 
